View source for Complete algebraic variety

A generalization of the concept of a compact complex algebraic variety. A separated variety $X$ is called complete if for any variety $Y$ the projection $X \times Y \rightarrow Y$  is a closed morphism, i.e. it maps closed subsets of $X \times Y$ (in the [[Zariski topology]]) into closed subsets of $Y$. 

Any [[projective variety]] is complete, but not vice versa. For any complete algebraic variety $X$ there exists a projective variety $X_1$ and a projective birational morphism $X_1\rightarrow X$ (Chow's lemma). For any algebraic variety $X$ there exists an open imbedding into a complete variety $\tilde X$ (Nagata's theorem). A generalization of the concept of a complete algebraic variety to the relative case is that of a [[proper morphism]] of schemes.

There is also the ''valuative completeness criterion'': For any discrete valuation ring $A$ with [[field of fractions]] $K$ and any morphism $u : \mathrm{Spec}\,K \rightarrow X$ there should be a unique morphism $v : \mathrm{Spec}\,A \rightarrow X$ that extends $v$. This condition is an analogue of the requirement that any sequence in $X$ has a limit point.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977)  {{ISBN|0-387-90244-9}} {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
</table>

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